# Performance Prediction Model for Hydrodynamically Lubricated Tilting Pad Thrust Bearings Operating under Incomplete Oil Film with the Combination of Numerical and Machine-Learning Techniques

^{*}

## Abstract

**:**

## 1. Introduction

_{2}− R

_{1})/R

_{2}= ½ were the optimum geometry under starved conditions. Finally, Artiles and Heshmat [11] performed an analysis on starved thrust bearings that included temperature effects. They used a finite difference mesh in order to solve the 2-D temperature and pressure fields. The investigation was performed for tapered land thrust bearings for different minimum film thicknesses and levels of starvation. It was found that the effects of starvation were small when the bearing was flooded with lubricant, but accelerated rapidly below 50% of starvation level. The start of the film was mainly independent of geometric characteristics, but directly dependent on the starvation level.

## 2. Theory

#### 2.1. Hydrodynamic Lubrication Model

#### 2.2. Viscosity Model

_{0}is considered to be the inlet temperature. The constant k

_{e}is empirical and, with a value of 0.8, gives good agreement between theory and experiment. The variation of temperature ΔT is considered to be a function of friction, rotating velocity and average axial fluid flow. The lubricant’s density and specific heat capacity are also taken into consideration. To add to that, the fraction $\frac{{l}_{in}}{L}$ is applied, in order to define the various percentages of inlet oil coverage during the investigation. An iterative procedure is followed, in order to define the final average effective temperature for each simulation.

#### 2.3. Numerical Analysis

_{atm}is applied as a boundary condition. To add to that, an outflow condition is prescribed in both inner and outer pad sides: r = R

_{in}, R

_{out}. In addition, no inflow is allowed in the computational domain and the ambient pressure P = p

_{atm}is applied. The rotor is assumed to be moving with a constant rotational velocity ω, which corresponds to U = ωr

_{mean}at the pad’s mid sector. An iterative algorithm is built based on the finite differences—central differences—methodology. The Reynolds equation is adapted so that the algorithm is able to swipe over the grid and compute the corresponding pressure P

_{ij}at any internal node (9). A representation of the calculation is presented in Figure 3, where c is the node at which the pressure is calculated and n, w, s, e are the neighboring nodes used for this calculation. Convergence to steady-state condition is verified by monitoring the computed nodal pressure based on the defined convergence criteria (10). In the cases of incomplete oil film (Figure 4), the lubricant’s width limit lines LB (i), LT (i) are calculated by swiping over the nodes in the direction of the flow (11). The amount of lubricant that enters the domain l

_{in}flows through the pad-rotor conjunction and adapts to the inclination of the pad. As a result, the same amount of lubricant at every step of the way through the pad (i) has to cover more and more of its surface until (if) it reaches the pad’s sides or the end of the pad in the flow direction. Pressure P = p

_{atm}is then applied as a boundary condition on the area where no lubricant flows. The calculation of pressure distribution in the y-direction is then limited to the new boundary conditions. In addition, Case A refers to lack of lubricant on the outer part of the pad, and is modeled with LB (i) placed on the inner pad border, while LT (i) takes values within the domain. Case B refers to the lack of lubricant on the inner part of the pad. As a result, LT (i) is placed on the outer border and LB (i) runs through the fluid film domain. Finally, Case C refers to the scenario where both LB (i) and LT (i) are calculated symmetrically through the fluid film.

#### 2.4. Machine-Learning

_{1i}: rotational velocity [rpm]; x

_{2i}: percentage of inlet oil coverage, in order to predict the response values of one dependent variable Y: Pad’s Load-carrying Capacity [N]. For a set of n-observations, Equation (12) or, in matrix form, Equation (13), is solved, in order to calculate the y-intercept: β

_{0}and the corresponding slopes: β

_{1},…,β

_{5}.

_{1i}and x

_{2i}. Then, all the mean squared errors were calculated separately for all the response values of both predictors (15) in each splitting candidate node t. At every iteration, the splitting node t of the regression tree was defined as the one that provided the minimum mean-squared error from all the examined data. The procedure continues repeatedly until each branch reaches the pre-defined leaf size. For the current study, a leaf size equal to 4 has been selected, as it provides the finest tree results for the Matlab’s application with the optimum accuracy. In addition, the criteria chosen in the current study, in order to measure and evaluate the goodness of fit for the generated machine-learning models, is the coefficient of determination, or R

^{2}(16). This coefficient indicates the difference between the values of the dependent variable y

_{fit}calculated from the model and the observations y

_{num}obtained from the relevant numerical simulations. The higher the value of R

^{2}, the better the model is at predicting the data. Finally, the Matlab’s standard 5-fold, cross-validation procedure was applied for 5 randomly chosen partitions of the original data set. All the models where trained with 80% of the data from the data lake, while the rest 20% of the data was used for testing. Experimental data were used for the validation of the ML model as shown in [16].

## 3. Results

^{2}values of each case:

^{2}values in all models are close to 0.99, which means that there is a good agreement between the numerical data and the prediction models’ response values. At the same time, this is also an indicator of 99% accuracy for the ML model to predict the pad’s load-carrying capacity at the given predictor values.

^{2}values, which will define the goodness of fit for all the trained models, are presented in Table 3. First of all, values of the order of 0.95 for the R

^{2}are, in general, accepted as very good for the fitness of the models in the data. That means that all trained models in this study have a very good response and higher than 95% accuracy to predict the load-carrying capacity of the pad. Nevertheless, in a more detailed approach, the Quadratic SVM models show better results than Regression Trees, while the Quadratic Polynomial Regression models present, in general, the best values of R

^{2}.

^{2}= 0.98, while the Regression Tree model has an R

^{2}= 0.95, which gives 4% less accuracy in load-carrying capacity prediction compared to the Quadratic Polynomial Regression model. Figure 13 is a graphical representation of the predicted versus the true response values for the Quadratic SVM and the Regression Tree models that were trained with Matlab’s Regression Learner tool. It is visually verified that the Quadratic SVM model has a better fit to the results compared to the Regression Tree model, since the observations (blue markers) are gathered very close to the prediction line compared to the Regression Tree model on the right, which shows a few observations with a higher deviation from the prediction line, mainly on the upper left corner. Figure 14 and Figure 15 (below) are the typical representation of the response plots for the SAE 10W40, case study C and Quadratic SVM model for each predictor. Similarly, Figure 16 and Figure 17 are the typical representations of the response plots for SAE 10W40, case study C and Regression Tree model. Finally, Figure 18 is the graphical representation of the Regression Tree machine-learning model for the lubricant SAE 10W40 and case study C- symmetrical, incomplete oil film profile.

## 4. Conclusions

- ▪
- As less oil covers the pad’s surface, the load-carrying capacity drops up to 93% for 40% of inlet oil coverage.
- ▪
- The load-carrying capacity of the pad is affected by the position of the oil film incompleteness. The lack of lubricant on the outer area of the pad, profile A, shows the worst load-carrying capacity results, while the case study C profile, with symmetrical lack of lubricant, presents up to 15% better performance.
- ▪
- From the studied lubricants, SAE 10W40 shows up to 135% better performance for the worst studied conditions of 12,000 rpm and 40% inlet oil coverage.
- ▪
- All the machine-learning models have a good accuracy in predicting the load-carrying capacity of the pad, since all R
^{2}values are higher than 0.95. - ▪
- Finally, the Quadratic Polynomial Regression ML model shows 1% better accuracy compared to the Quadratic SVM model, and 4% better accuracy when compared to the Regression Tree ML model.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

$A$ | total area of bearing pads [m^{2}] |

$B$ | pad length in x-direction [m] |

${C}_{1}^{\mu}$ | first viscosity coefficient—absolute temperature at which μ = μ_{ν} (323 K) |

${C}_{2}^{\mu}$ | second viscosity coefficient according to Sutherland’s law = 3800 |

${C}_{3}^{\mu}$ | third viscosity coefficient according to Sutherland’s law = 30,000 |

${C}_{n,s,w,e}$ | constants for each neighbor node |

$h$ | film thickness [m] |

${h}_{0},{h}_{1}$ | outlet, inlet film thickness [m] |

${h}_{min}$ | minimum film thickness [m]: ${h}_{min}=\mathrm{min}\left({h}_{0},{h}_{1}\right)$ |

$k$ | convergence ratio: $k=\left({h}_{1}-{h}_{0}\right)/{h}_{0}$ |

${k}_{e}$ | empirical constant = 0.8 [21] |

$L$ | pad’s width in y-direction [m] |

$p$ | absolute pressure [Pa] |

$P$ | absolute nodal pressure [Pa] |

${q}_{x,y}$ | lubricant flow [m^{3}/h] |

${Q}_{in,out}$ | lubricant flow in inlet and outlet area of the pad [m^{3}/h] |

${Q}_{sr1,2}$ | lubricant outflow from the sides of the pad [m^{3}/h] |

$T$ | temperature [K] |

$U$ | linear rotor velocity [m/s] |

$\mu $ | dynamic viscosity coefficient [Pas] |

${\mu}_{v}$ | nominal dynamic viscosity |

$x$ | independent variable of length along pad’s width side [m] |

$\omega $ | rotational velocity [rpm] |

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**Figure 6.**Typical pad’s pressure distribution for the Case A incomplete oil film profile at 60% oil film coverage for the inlet of the pad.

**Figure 7.**Typical pad’s pressure distribution for the Case B incomplete oil film profile at 60% oil film coverage for the inlet of the pad.

**Figure 8.**Typical pad’s pressure distribution for the Case C incomplete oil film profile at 60% oil film coverage for the inlet of the pad.

**Figure 9.**Quadratic Polynomial Regression model of SAE30 for all the incomplete oil film profiles. Load-carrying capacity according to percentage of inlet oil coverage and rotational velocity.

**Figure 10.**Quadratic Polynomial Regression model of SAE10W40 for all the incomplete oil film profiles. Load-carrying capacity according to percentage of inlet oil coverage and rotational velocity.

**Figure 11.**Quadratic Polynomial Regression model of SAE 20 for all the incomplete oil film profiles. Load-carrying capacity according to percentage of inlet oil coverage and rotational velocity.

**Figure 12.**Quadratic Polynomial Regression model of incomplete oil film profile C for all the studied lubricants. Load-carrying capacity according to percentage of inlet oil coverage and rotational velocity.

**Figure 13.**SVM model VS Regression Tree model- True and Prediction response plots for SAE 10W40, case C.

**Figure 14.**Typical response plot of the pad’s inlet oil coverage and load-carrying capacity for the Quadratic SVM model, SAE 10W40, case C profile.

**Figure 15.**Typical response plot of the rotational velocity and load-carrying capacity for the Quadratic SVM model, SAE 10W40, case C profile.

**Figure 16.**Typical response plot of the pad’s inlet oil coverage and load-carrying capacity for the Regression Tree model, SAE 10W40, case C profile.

**Figure 17.**Typical response plot of the pad’s rotational velocity and load-carrying capacity for the Regression Tree model, SAE 10W40, case C profile.

**Figure 18.**Graphical representation of the Regression Tree model for the SAE 10W40, case C incomplete oil film profile.

Pad’s Length | 32 | mm |

Pad’s Width | 28 | mm |

Pad’s Outer Radious | 62 | mm |

Pad’s Inclination | 0.1 | |

Pad’s Pivot | center | |

Rotational Velocity | 2000–12,000 | rpm |

Percentage of Inlet Oil Coverage | 0.4–1 | |

SAE 20 dynamic viscosity @50 °C | 0.033 | Pasec |

SAE 30 dynamic viscosity @50 °C | 0.046 | Pasec |

SAE 10W40 dynamic viscosity @50 °C | 0.054 | Pasec |

SAE 20 density @40 °C | 861 | Kg/m^{3} |

SAE 20 specific heat capacity | 2021 | J/kgK |

SAE 30 density @40 °C | 869 | Kg/m^{3} |

SAE 30 specific heat capacity | 1950 | J/kgK |

SAE 10W40 density @40 °C | 851 | Kg/m^{3} |

SAE 10W40 specific heat capacity | 1980 | J/kgK |

Lubricant’s Inlet Temperature | 323 | K |

Case Study | ML Model | R^{2} |
---|---|---|

SAE 30 Case A | $y=139.4-891{x}_{1}-0.016{x}_{2}+1577{x}_{1}{}^{2}+0.075{x}_{1}{x}_{2}-0.1\times {10}^{-5}{x}_{2}{}^{2}$ | 0.99 |

SAE 30 Case B | $y=5.1-405.3{x}_{1}-0.021{x}_{2}+1240.7{x}_{1}{}^{2}+0.08{x}_{1}{x}_{2}-0.8\times {10}^{-6}{x}_{2}{}^{2}$ | 0.99 |

SAE 30 Case C | $y=-57.7-189.7{x}_{1}-0.02{x}_{2}+1087.3{x}_{1}{}^{2}+0.08{x}_{1}{x}_{2}-0.8\times {10}^{-6}{x}_{2}{}^{2}$ | 0.99 |

SAE 10W40 Case A | $y=172.3-1035.4{x}_{1}-0.023{x}_{2}+1792.5{x}_{1}{}^{2}+0.09{x}_{1}{x}_{2}-0.1\times {10}^{-5}{x}_{2}{}^{2}$ | 0.99 |

SAE 10W40 Case B | $y=101.7-748.3{x}_{1}-0.026{x}_{2}+1593.4{x}_{1}{}^{2}+0.09{x}_{1}{x}_{2}-0.8\times {10}^{-6}{x}_{2}{}^{2}$ | 0.99 |

SAE 10W40 Case C | $y=23.1-496.5{x}_{1}-0.023{x}_{2}+1419.2{x}_{1}{}^{2}+0.09{x}_{2}-0.9\times {10}^{-6}{x}_{2}{}^{2}$ | 0.99 |

SAE 20 Case A | $y=80.3-729.3{x}_{1}-0.01{x}_{2}+1409.1{x}_{1}{}^{2}+0.07{x}_{1}{x}_{2}-0.1\times {10}^{-5}{x}_{2}{}^{2}$ | 0.99 |

SAE 20 Case B | $y=-38.7-325.4{x}_{1}-0.009{x}_{2}+1127.1{x}_{1}{}^{2}+0.07{x}_{1}{x}_{2}-0.1\times {10}^{-5}{x}_{2}{}^{2}$ | 0.99 |

SAE 20 Case C | $y=-909-1418.2{x}_{1}-0.09{x}_{2}+9977.2{x}_{1}{}^{2}+0.7{x}_{1}{x}_{2}-0.1\times {10}^{-4}{x}_{2}{}^{2}$ | 0.99 |

Case Study | R^{2} |

SAE 30 Quadratic SVM ML model | 0.98 |

SAE 30 Regression Tree ML model | 0.95 |

SAE 10W40 Quadratic SVM ML model | 0.98 |

SAE 10W40 Regression Tree ML model | 0.95 |

SAE 20 Quadratic SVM ML model | 0.98 |

SAE 20 Regression Tree ML model | 0.95 |

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## Share and Cite

**MDPI and ACS Style**

Katsaros, K.P.; Nikolakopoulos, P.G.
Performance Prediction Model for Hydrodynamically Lubricated Tilting Pad Thrust Bearings Operating under Incomplete Oil Film with the Combination of Numerical and Machine-Learning Techniques. *Lubricants* **2023**, *11*, 113.
https://doi.org/10.3390/lubricants11030113

**AMA Style**

Katsaros KP, Nikolakopoulos PG.
Performance Prediction Model for Hydrodynamically Lubricated Tilting Pad Thrust Bearings Operating under Incomplete Oil Film with the Combination of Numerical and Machine-Learning Techniques. *Lubricants*. 2023; 11(3):113.
https://doi.org/10.3390/lubricants11030113

**Chicago/Turabian Style**

Katsaros, Konstantinos P., and Pantelis G. Nikolakopoulos.
2023. "Performance Prediction Model for Hydrodynamically Lubricated Tilting Pad Thrust Bearings Operating under Incomplete Oil Film with the Combination of Numerical and Machine-Learning Techniques" *Lubricants* 11, no. 3: 113.
https://doi.org/10.3390/lubricants11030113